tag:blogger.com,1999:blog-3891434218564545511.post7109733239012166647..comments2021-05-17T15:45:35.218-05:00Comments on Alexander Pruss's Blog: The Axiom of Choice in some claims about probabilitiesAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-3891434218564545511.post-37850439925478525662013-11-28T09:06:57.455-06:002013-11-28T09:06:57.455-06:00In the original version of the post I said that I ...In the original version of the post I said that I thought I could prove that Hahn-Banach implies 1. But my proof is mistaken. Oops.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-19746024775584722122013-11-24T12:42:03.140-06:002013-11-24T12:42:03.140-06:00Here's a sketch of the proof of 2 from HB:
By...Here's a sketch of the proof of 2 from HB:<br /><br />By the ideal-filter duality, HB is equivalent to the claim for any boolean algebra and filter, there is a probability measure that makes every member of the filter have unit probability.<br /><br />Let B be any boolean algebra. Let V be the set of all non-empty finite boolean subalgebras of B. For a in V, let a* be the filter { b in V : a subseteq b }. Let F be the filter on 2^V generated by all the a*. By Hahn-Banach, let mu be a measure on 2^V that makes every member of F have unit probability. Let 'R = R^V / ~, where f~g iff mu{ f = g } = 1.<br /><br />Finally, for each U in V, let nu_U be the probability measure on the boolean algebra U that assigns equal weight to each atom of U. <br /><br />Define an 'R-valued measure nu on B. Fix a in B. Then nu(a) is supposed to be an equivalence class of functions from V to R. We shall define a representative f_a in R^V. Fix U in V. If a is not in U, then let f_a(a)=1. If a is in U, then let f_a(a)=nu_U(a). Let nu(a) = [f_a]. Since {a}* in F, mu({a}*) = 1, so we can ignore the case where a is not in U for verifying additivity.<br /><br />Write x < y for x and y in 'R iff mu { x < y } = 1. If 0 < y, then we can divide by y.<br /><br />Define st:'R -> R cup {UND} (for any element UND not in R) by st([f]) = I_mu(f) if f is mu-almost surely bounded, where I_mu is the integral of f over V with respect to mu, and st([f]) = UND otherwise.<br /><br />Note that nu(a) > 0 for any nonzero a in B, since f_a > 0 on {a}* and mu({a}*}=1. Thus, we can define<br />P(a|b) = st( mu(ab)/mu(b) ).Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-29270639518372509232013-11-24T12:27:23.025-06:002013-11-24T12:27:23.025-06:00What I am currently most curious about is whether ...What I am currently most curious about is whether my (2) implies HB. <br /><br />By the way, it would be interesting to see if some of the ways of translating pointless topology to topology could not be adapted to use HB instead of BPI, basically replacing ultrafilters with probability measures. (Here's what makes me wonder about this. A couple of days ago, I noticed that given HB, you can represent any boolean algebra as closed sets in a compact topological space (just represent A by all the probability measures with support in A). The embedding preserves meets but doesn't preserve negations. So you get a part of what BPI gives you with HB.)Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-28198770006550304852013-11-24T12:06:54.581-06:002013-11-24T12:06:54.581-06:00Thanks! I knew that BPI→HB is proper, but I didn&#...Thanks! I knew that BPI→HB is proper, but I didn't know that BT→Lebesgue Nonmeasurable Sets was. That's nice to know. <br /><br />HB→BT seems to be proper (see <a href="http://consequences.emich.edu/CONSEQ.HTM" rel="nofollow">here</a> and put in 309 to be true and 52 to be false), which is unsurprising.<br /><br />I haven't done anything with topoi since I was teenager. :-) Might be good to look at them again.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-945422666677268172013-11-24T11:12:22.855-06:002013-11-24T11:12:22.855-06:00Just in case it is useful:
According to Herrlich...Just in case it is useful: <br /><br />According to Herrlich, "Axiom of Choice" pg. 134, diagram 5.25, we have<br /><br />BPI => HB => Banach-Tarski => Non-Lebesgue measurable sets exist.<br /><br />All the implications are proper with the possible exception of HB => Banach-Tarski.<br /><br />Herrlich defers to Howard and Rubin, "Consequences of the axiom of choice" for some of the details (mostly, those that have to do with the construction of ZF models where one theorem or other does not hold) which is the standard monograph to check for this kind of stuff. There is website associated to the book where you can fill out forms and get back such info as all the equivalent versions of some choice principle and such like. The link is <a href="http://www.math.purdue.edu/~hrubin/JeanRubin/Papers/conseq.html" rel="nofollow">here</a>.<br /><br />A couple more random, potentially interesting notes. <br /><br />(1) By Solovay and Shelah's results, ZF + AC + IC (existence of an inaccessible cardinal) is equiconsistent with ZF + DC (dependent choice) + LM (all subsets of R are Lebesgue measurable).<br /><br />(2) BPI is equivalent to Tychonoff for compact Hausdorff spaces. The latter has a constructively valid proof; the catch is that the proof is made in the world of locales and to translate the proof to topological spaces, BPI is needed. Johnstone's book "Stone Spaces" is the go-to monograph for this kind of "pointless topology" (Johnstone himself wrote a paper whimsically titled "The point of pointless topology"). HBT, as some other analysis big guns like Gelfand-Naimark duality, itself have constructively valid versions and proofs in the context of topoi. Measure theory can also be done internally, and thus constructively, to a topos. This is an area of active research, but a starting reference is Mathew Jackson's phd thesis <a href="http://www.andrew.cmu.edu/user/awodey/students/jackson.pdf" rel="nofollow">A sheaf theoretic approach to measure theory</a>. Some of this stuff is "just" internalizing to a topos constructions with measure *algebras* (not spaces), for which the reference is Fremlin's vol.3 in his 5-volume opus dedicated to measure theory, also available online.grodrigueshttps://www.blogger.com/profile/12366931909873380710noreply@blogger.com